Root finding using Halley's rational method (the second in
the class of Householder methods). Assumes the function is three
times continuously differentiable and converges cubically when
progress can be made.

AD serves as a common wrapper for different Mode instances, exposing a traditional
numerical tower. Universal quantification is used to limit the actions in user code to
machinery that will return the same answers under all AD modes, allowing us to use modes
interchangeably as both the type level "brand" and dictionary, providing a common API.